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Full text of "Treatise On Analysis Vol-Ii"

328       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

may assume that  \fd\L = 1, and then there exists a sequence (un) with the properties
above. Use the relation Di(k/| • ju,) = Wk/l) to deduce that the sequence of
integrals f k/|1/2 du tends to 1, and use Problem 4 of Section 13.12.)
(b)   Let /e E(/z), and let g e JS?°°(/i) be such that fg e K(/z). Show that

and that  (fgd\L-= \[fdp\[ (g d\j\. (Consider a sequence (un) in B such that

unf^ 1 in the mean, and for each v e B consider the integrals Junvfg d^.)

(c)   If/, g in E(fji) are such that |/(jc)| = \g(x)\ almost everywhere, then/= eg almost
everywhere, where c is a constant. (Use (b) to show that the function fig belongs to

(d)    For each function /eK(/x) such that JWI/I) > 0, show that f=gh, where
# e E(ji) and h e I(ju,), and that g, h are unique up to a constant factor of absolute
value 1. (Use Problem 12(c), and (b) and (c) above.)

(e)   If /, g belongs to K(f>0 n -S?°V), then J /# rf/x = (Jfdp,j (j # 4tj. (Notice first

that if /z e K(^) n &°°(/x) is such that J/z^t = 0, then J h2 dp = 0. To prove this,
use the inequality

1 = J (1 + zh) dfji

valid for all z e C, and deduce that 1 ^ I 11 — z2h2\ d\i\ then use Problem 7 of Section

13.12. To deal with the general case, consider the function h =/-f zg — I (/+ zg) dp,

where z e C.)

(f)   If /, g in K(/x) are such that fg e ^(p), show that fg e K(/j,) and that

(Assume first that g e £(/*), and show that, for each u E K(ft) such that ug e

we have DI(|#| • /i)   |w dp\ ^ I |w^| J/u-. Deduce as in Problem 10(d) that there exists

a function ve -Sf°°(ft) such that \v\ ^ 1 and DI(|^| • /-i) \udfji= \ugvdyi for all

H e K(//,) such that ug e &l(p). Using (b), conclude that u is a constant of absolute
value 1. In the general case, show first that, by addition of a suitable constant to

/and g, we may assume that \fdp. ^ 0 and I g dp =£ 0. Then use the canonical de-
composition of (d) above, and show with the help of (e) that if/and g are in !(/*)
then so isfg.)

(g)   Deduce from (f) that if/, #eE(/z) are such that^ e -fi?V)> then^ e E(^).
(h)   If/e JS?1^) and gEE(fjC) and /#eK(/x), show that /e K(/x). (By adding

a suitable constant to /, reduce to the case where Ifgdp^Q, and deduce that
0. Then show, by using the results above, that /= uv withtative measure for x- Then DI(/- p.) == J^(/) for all/ ^ 0 in &l(p) (Problem ll(b)).